We consider a model of a circular lenticular vortex immersed into a deep and vertically stratified viscous fluid in the presence of gravity and rotation. The vortex is assumed to be baroclinic with a Gaussian profile of angular velocity both in the radial and axial directions. Assuming the base state to be in cyclogeostrophic balance, we derive linearized equations of motion and seek for their solution in a geometric optics approximation to find amplitude transport equations that yield a comprehensive dispersion relation. Applying the algebraic Bilharz criterion to the latter, we establish that the stability conditions are reduced to three inequalities that define the stability domain in the space of parameters. The main destabilization mechanism is either monotonic or oscillatory axisymmetric instability depending on the Schmidt number (Sc), vortex Rossby number, and the difference between radial and axial density gradients as well as the difference between epicyclic and vertical oscillation frequencies. We discover that the boundaries of the regions of monotonic and oscillatory axisymmetric instabilities meet at a codimension-2 point, forming a singularity of the neutral stability curve. We give an exhaustive classification of the geometry of the stability boundary, depending on the values of the Schmidt number. Although we demonstrate that the centrifugally stable (unstable) Gaussian lens can be destabilized (stabilized) by the differential diffusion of mass and momentum and that destabilization can happen even in the limit of vanishing diffusion, we also describe explicitly a set of parameters in which the Gaussian lens is stable for all Sc >0.

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